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arX
iv:p
hysi
cs/0
1120
89v1
[ph
ysic
s.ch
em-p
h] 2
6 D
ec 2
001
QED-SCF, MCSCF and Coupled-cluster Methods in Quantum
Chemistry
Tadafumi Ohsaku∗ and Kizashi Yamaguchi
Department of Physics and Department of Chemistry, Graduate School of Science, Osaka
University, Machikaneyama-cho 1-1, Toyonaka, Osaka, 560-0043, Japan
Abstract
We investigate the method to combine the techniques of quantum chemisty
with QED. In our theory, we treat the N-electron system and the Dirac sea on
an equal footing; we regard both of them as the dynamical degrees of freedom
of a many-body system. After the introduction of our QED-SCF method, the
QED-SCF solutions are classified into several classes on the basis of group-
theoretical operations such as time reversal, parity and O(3) rotational sym-
metry. The natural orbitals of general QED-SCF solutions are determined
by diagonalizing the first order density matrix. Thus, we obtain the possibil-
ity to treat the system under strong QED effect by the methods of quantum
chemistry, such as QED-MCSCF and QED-coupled-cluster approaches.
keywords; QED, QED-SCF, MCSCF, Coupled-cluster theory, Radiation chemistry.
I. INTRODUCTION
The Hartree-Fock ( HF ) and multiconfiguration HF ( MCHF ) methods have been well-
established for nonrelativistic Hamiltonian, and the self-consistent field ( SCF ) equations
∗Corresponding author; Tadafumi Ohsaku, [email protected]
1
to obtain the HF and MCHF solutions are also derived in several ways [1]. The most general
HF solution for the nonrelativistic case is given by the general spin orbitals ( GSO ): The
two-component spinor [2,3],
φi = φ+i α + φ−
i β =
ψ+i
ψ−i
. (1)
Very recently we have developed the program package for ab initio HF and density functional
( DFT ) calculations of molecules by using the GSO [4,5]. It was shown that GSO HF
and GSO DFT approaches are useful for qualitative understanding of correlation and spin
correlation effects in polyradicals with orbital degeneracies [4,5].
By the way, it is a well-known fact that, in atomic and moleculer physics, the relativistic
effects become crucial in many cases. Especially in the electronic structure calculation for
heavy elements, we have to introduce a kind of relativistic treatment. For example, the case
of neutral heavy elements can be treated satisfactorily by the Dirac-Coulomb-Breit no-sea
scheme [6]. This method is based on the no-sea (the approximation completely neglects
the effects of the Dirac sea) Hamiltonian. It has a Dirac-type one-body Hamiltonian, and
the interaction of electrons is treated via the Coulomb potential and Breit operator [7].
This Hamiltonian with several many-body schemes (HF, relativistic many-body perturbation
theory ( RMBPT ), so on) give good numerical results for neutral heavy elements in atoms
and molecules. But, especially in the case of highly charged heavy elements, the accurate
second-order RMBPT indicates that quantum electrodynamical ( QED ) corrections are
clearly seen in the difference between RMBPT and experiment. Various relativistic schemes
have already been presented in the electronic structure theory, and they are summarized in
recent review articles [6,8,9].
In the previous paper [10], we constructed the theory of quantum electrodynamical self-
consistent fields ( QED-SCF ). We derived the time-dependent Hartree-Fock ( TDHF )
theory, HF condition and random phase approximation ( RPA ) [11,12]. Because we con-
structed our theory based on QED, we regard them as an intrinsic treatment of relativistic
theory in atomic and moleculer physics. In this theory, we treat an N-electron system and
2
the Dirac sea on an equal footing, and they interact by exchanging photons. Thus our theory
can treat the QED effects such as the vacuum polarization effect, which will be observed
around the nucleus of heavy element ( strong external Coulomb field ) [8,9,13]. To describe
the QED effects, we have to treat not only N-electron system but also the Dirac sea as the
dynamical degrees of freedom in a many-body system. Futhermore, because we gain the
ability to introduce various many-body techniques of atoms and molecules to QED, we can
discuss the availabilities of post-HF scheme through consideration of the QED-HF stability
condition, like the GSO model in the nonrelativistic theory.
This paper is organized as follows. In Sec. II, we give a brief summary of our method to
combine many-body techniques to QED. In Sec. III, we perform group-theoretical character-
izations of QED-SCF solutions [10] on the basis of time-reversal, parity and O(3) rotational
symmetry. Under these group-operations, the first-order density matrix of these solutions
is classified into several different classes. In Sec. IV, we give some examples to apply our
method to quantum chemistry. We examine the post QED-SCF methods as in the case of
post HF methods in quantum chemistry. The MBPT, coupled-cluster ( CC ) and MCSCF
methods are considered in the QED scheme. To this end, we utilize previous procedures
for the nonrelativistic case [14]: (1) Use of density matrix, (2) its diagonalization to obtain
the natural orbitals given by four-component bispinor, and (3) selection of active space for
MCSCF. Finally in Sec. V, we give discussion about this work.
II. QED-HF THEORY
In this section, we give a brief summary of our QED-SCF method which has been inves-
tigated in previous paper [10]. Our QED-SCF method is based on three factors: First, we
introduce a QED Hamiltonian written by products of the one-particle operators ( creation-
annihilation operators ). Second, we introduce an assumption for evaluation of matrix
elements. Third, we use the Thouless form for the determinantal state [11,15]. By intro-
ducing these three tools, it becomes possible to combine various many-body techniques with
3
QED.
A. Hamiltonian
We start from the following Lagrangian
L(x) = −1
4Fµν(x)F
µν(x) + ψ(x)(iγµ∂µ −m0)ψ(x) + eψ(x)γµψ(x)(Aµ(x) + A(e)µ (x)). (2)
Here, Aµ is a ( virtual ) photon field which describes the interaction between fermions, and
A(e)µ is a classical external field. Next we remove the photon degree of freedom and introduce
a Coulomb potential U(r) of a nucleus ( or nuclei ) as the external field. Then we obtain
the following Hamiltonian:
H =∫
d3x ˆψ(x)(−i~γ · ∇+m0 + γ0U(r))ψ(x)
+1
2e2∫
d3x∫
d4y ˆψ(x)γµψ(x)Dµν(x− y)ˆψ(y)γνψ(y). (3)
Here each field is quantized. Dµν(x − y) is the full photon propagator. The γ-matrices are
defined by ( under the standard representation )
γ0 =
I 0
0 −I
, γi =
0 σi
−σi 0
, (4)
where I is the unit matrix, and σi ( i=1,2,3 ) is the Pauli matrix:
σ1 =
0 1
1 0
, σ2 =
0 −i
i 0
, σ3 =
1 0
0 −1
. (5)
In the above Hamiltonian, we expand each field operator by one-particle state function as
ψ(x) =∑
i
(ψ(+)i (x)ai + ψ
(−)i (x)b†i ), (6)
ˆψ(x) =∑
i
(ψ(+)i (x)a†i + ψ
(−)i (x)bi), (7)
where +(−) means the electron ( positron ) state, i denotes quantum numbers for one-
particle states, a†i (ai) is electron creation ( annihilation ) operator, b†i (bi) is positron creation
( annihilation ) operator, and ψ(±)i is the four-component bispinor given like
4
ψ(±)i = (χ
(±)(1)i , χ
(±)(2)i , χ
(±)(3)i , χ
(±)(4)i ). (8)
Then we obtain the Hamiltonian written by creation-annihilation: operators
H = HK + HI , (9)
HK =∑
i,j
T±±ij (a†i aj + a†i b
†j + biaj − b†j bi), (10)
HI =1
2
∑
i,j,k,l
V ±±±±ijkl (a†i a
†kalaj + a†i a
†k b
†l aj + a†i aj bkal + a†i b
†l aj bk
+a†i b†j a
†kal + a†i b
†j a
†k b
†l + a†i b
†j bkal + a†i b
†l b
†j bk + a†k biajal + a†k b
†l biaj
+biaj bkal + b†l aj bibk + b†j a†k bial + a†k b
†j b
†l bi + b†j bk bial + b†j b
†l bk bi), (11)
where we designate matrix elements as
T±±ij =
∫
d3xψ(±)i (x)(−i~γ · ∇+m0 + γ0U(r))ψ
(±)j (x), (12)
V ±±±±ijkl =
1
2e2∫
d3x∫
d3yψ(±)i (x)γµψ
(±)j (x)
gµν exp(i∆ǫ|x− y|)
4π|x− y|ψ
(±)k (y)γνψ
(±)l (y).
(13)
Here, ∆ǫ = |ǫk − ǫl| = |ǫi − ǫj | ( the energy difference of one-particle states ). gµν is
usual metric tensor and defined as gµν = diag(1,−1,−1,−1). About the interaction, we
only take into account 0th order bare photon propagator, and choose the Feynman gauge
for convenience. This Hamiltonian can describes the QED effects such as the vacuum-
polarization, because which includes the Dirac sea as the dynamical degree of freedom.
Hereafter we use this Hamiltonian in our theory.
Next we give the method for evaluation of expectation value. In QED, the Dirac vacuum
under the presence of the external Coulomb field generates 4-current as an observed effect,
which is called the vacuum polarization [8,9,13]. It is well-known that, this effect is described
by the charge conjugation symmetric 4-current given in the next form [8,9,13]
jvacµ (x) = −e
2
(
∑
n<−m
ψ(−)n (x)γµψ
(−)n (x)−
∑
n>−m
ψ(+)n (x)γµψ
(+)n (x)
)
. (14)
5
This expression gives the 4-current induced by the external Coulomb field. In this expression,
in the external-field-free case two sums cancel, but for the field present case two sums do
not cancel completely. Hereby, this result should be interpreted as describing the vacuum
polarization effect which is induced by the external Coulomb field. Therefore, the current
of N-electron system with adding the vacuum polarization ( here we consider the case that
the N-electrons occupy the one-particle states up to the Fermi level, as illustrated in Fig. 1
) is given as
jµ(x) = −e∑
−m<i≤F
ψ(+)i (x)γµψ
(+)i (x)
−e
2
(
∑
i<−m
ψ(−)i (x)γµψ
(−)i (x)−
∑
i>−m
ψ(+)i (x)γµψ
(+)i (x)
)
. (15)
The second line gives the vacuum polarization [8,9,13]. To include this effect in our Hamil-
tonian, we have to modify the contraction scheme. From the definition of the contraction:
A•(t)B•(t′) = 〈F |A(t)B(t′)|F 〉θ(t− t′)− 〈F |B(t′)A(t)|F 〉θ(t′ − t). (16)
Here, A(t) and B(t) are some kind of operators, and |F 〉 is a Fermi sea. We introduce the
definition for the step function at the same time as θ(t− t′)t=t′ = θ(t′ − t)t=t′ =12. Then we
obtain the following relation:
〈F |T (A(t)B(t′))|F 〉t=t′ = 〈F |A•(t)B•(t′)|F 〉t=t′
=1
2〈F |A(t)B(t)|F 〉 −
1
2〈F |B(t)A(t)|F 〉. (17)
Let us consider the fact that, it is clear from the Hamiltonian given in Eq. (3), the expecta-
tion value of it will be written in a functional of the 4-current jµ, as same as the discussion in
Ref. 8. Therefore, we introduce a hypothesis that our Hamiltonian is written by operators
in the same-time T -products, and when we factorize the vacuum expectation value for an
operator product with the aid of the Wick theorem, each contraction should be calculated
by the definition given in (17).
By using the above method, for example, the expectation value of H is given by
6
〈F |H|F 〉 =1
2
∑
i<−m
T−−ii +
1
2(∑
−m<i≤F
−∑
i>F
)T++ii
+1
8
{
(∑
−m<i≤F
−∑
i>F
)(∑
−m<k≤F
−∑
k>F
)(V ++++iikk − V ++++
ikki )
+(∑
−m<i≤F
−∑
i>F
)(∑
k<−m
)(V ++−−iikk − V +−−+
ikki )
+(∑
−m<k≤F
−∑
k>F
)(∑
i<−m
)(V −−++iikk − V −++−
ikki )
+(∑
i<−m
)(∑
k<−m
)(V −−−−iikk − V −−−−
ikki )}
. (18)
This gives the HF energy in our theory. We argue that we can generalize this method to
calculate expectation values for arbitrary products of operators. By using the Wick theorem,
we always factorize the matrix elements of an operator product into the sum of the products
of contractions, then the contraction is calculated by the definition given above. It is clear
from the above discussion that only the definition of the contraction for same-time operators
is modified.
B. HF solution
Now, to obtain the relativistic Slater determinant in the Thouless form [11], we will
obtain a relativistic exponential operator which transforms one to another representations.
We assume the following relations for the canonical transformation:
a′m = eiS1 ame−iS1 =
∑
n
(αmnan + βmnb†n), (19)
a′†m = eiS1 a†me−iS1 =
∑
n
(α∗mna
†n + β∗
mnbn), (20)
b′†m = eiS1 b†me−iS1 =
∑
n
(βmnan + γmnb†n), (21)
b′m = eiS1 bme−iS1 =
∑
n
(β∗mna
†n + γ∗mnbn), (22)
where S†1 = S1 will be satisfied. Here new operators are expanded by a complete set of the
old representation. We introduce the exponential operator in the following form:
eiS1 = exp{
i∑
mn
(αmna†man + βmna
†mb
†n + β∗
mnbnam + γ∗mnbnb†m)}
. (23)
7
Here we demand that the matrices formed by the paramerters αmn, βmn, β∗mn, γ
∗mn should be
Hermitian. Then the exponential operator formally given above is unitary. It is clear that
the operator (23) will give the relations in (19)∼(22). We will write a relativistic Slater
determinant as
|Φ(α, β, β∗, γ∗)〉 = eiS1(α,β,β∗,γ∗)|Φ0〉. (24)
Here the Slater determinant of old representation is expressed by
|Φ0〉 =∏
i
a†i |0〉 (25)
and the Dirac vacuum is defined by ai|0〉 = 0 and bi|0〉 = 0.
By using the relativistic Slater determinant given above, we can write the expectation
value of our Hamiltonian into an expanded form, by the same way as nonrelativistic cases.
The expectation value of our Hamiltonian is given as follows:
〈Φ(α, β, β∗, γ∗)|H|Φ(α, β, β∗, γ∗)〉 = 〈Φ0|H|Φ0〉
+i〈Φ0|[H, S1(α, β, β∗, γ∗)]|Φ0〉
+i2
2!〈Φ0|[[H, S1(α, β, β
∗, γ∗)], S1(α, β, β∗, γ∗)]|Φ0〉
+O(S31). (26)
In (26), the first line is the HF total energy, the second line gives the first derivatives with
respect to the parameters (α, β, β∗, γ∗), and it must be zero in the HF condition. The third
line corresponds to the second derivatives and they determine the stability of the HF state:
i2
2!〈Φ0|[[H, S1(α, β, β
∗, γ∗)], S1(α, β, β∗, γ∗)]|Φ0〉 ≥ 0, (27)
and it is equivalent to the stability of collective modes in the RPA. It is obvious fact that the
operator formalism makes possible to do these discussions in QED. We gave the evidence
in previous paper [10] that, by using our method, we can derive the TDHF equation, HF
condition and RPA equation in QED with no inconsistency. Thus we can obtain the most
stable generalized QED solution at the HF level.
8
III. GROUP-THEORETICAL CLASSIFICATION OF GENERALIZED QED-HF
SOLUTIONS
A. Group-theoretical classification
In this section, we give a brief discussion of group-theoretical classification of the gen-
eralized QED-HF solutions. We firstly introduce the density matrix like the work of Fuku-
tome [16]:
Q4×4 = −〈ψ(x)ψ(x)〉4×4. (28)
Here ψ and ψ are usual Dirac field, and they are 4-component bispinors. Thus our density
matrix is 4×4-matrix, as denoted above. We can expand the 4×4 density matrix into the
16-dimensional complete set of γ-matrices:
Q4×4 = QSI +QVµ γ
µ +QTµνσ
µν +QAµ γ5γ
µ +QP iγ5. (29)
In this expansion, we take a convention that S denotes the scalar, V denotes the vector,
T denotes the 2-rank antisymmetric tensor, A denotes the axial vector and P denotes the
pseudoscalar. I is the 4× 4 unit-matrix, γµ is usual Dirac gamma matrix, σµν is defined as
σµν = i2[γµ, γν ], and γ5 = iγ0γ1γ2γ3. Thus we obtain the Lorentz structure in our density
matrix Q.
In the relativistic theory, we usually treat the Poincare group ( 4-translation and the
Lorentz group ), C ( charge conjugation ), P ( parity ), T ( time-reversal ). Under the
charge conjugation, ψ and ψ are transformed as
ψ → CψT , ψ → −ψTC−1. (30)
Then the density matrix is transformed as
Q = −〈ψψ〉 → −C〈ψψ〉TC−1 = CQTC−1
= QSI −QVµ γ
µ −QTµνσ
µν +QAµγ5γ
µ +QP iγ5. (31)
9
Here C ≡ iγ2γ0 is the charge conjugation matrix, and T denotes the transposition of matrix.
Under the time reversal,
ψ(t) → Tψ(−t), ψ(t) → ψ(−t)T, (32)
together with the rule of taking the complex conjugate about c-numbers, we obtain
Q(t) = −〈ψ(t)ψ(t)〉 → −T 〈ψ(−t)ψ(−t)〉∗T = TQ(−t)∗T
= QS∗(−t)I +QV ∗0 (−t)γ0 −QV ∗
i (−t)γi +QT∗0i (−t)σ
0i
−QT∗ij (−t)σ
ij +QA∗0 (−t)γ5γ
0 −QA∗i (−t)γ5γ
i −QP∗(−t)iγ5. (33)
Here T ≡ iγ1γ3 and i, j = 1, 2, 3. Under the spatial inversion,
ψ(x) → γ0ψ(−x), ψ(x) → ψ(−x)γ0, (34)
the density matrix is transformed as
Q(x) = −〈ψ(x)ψ(x)〉 → −γ0〈ψ(−x)ψ(−x)〉γ0 = γ0Q(−x)γ0
= QS(−x)I +QV0 (−x)γ0 −QV
i (−x)γi −QT0i(−x)σ0i
+QTij(−x)σij −QA
0 (−x)γ5γ0 +QA
i (−x)γ5γi −QP (−x)iγ5. (35)
The QED-HF solutions can be group-theoretically classified into several types as in
the case of nonrelativistic HF solutions [16]. To consider this problem, we determine the
symmetry-group of a system. In the atomic or molecular systems, the translation invariance
is broken. In the case of an atom, only O(3) rotation is remained in the Lorentz group (
In the case of a molecule, O(3) is replaced by the point group.). Under the O(3) rotational
symmetry, we expand the QS, QP , QV0 and QA
0 by the scalar spherical harmonics, while we
expand the QVi , Q
Ai and QT
µν by the vector spherical harmonics [17]. It is clear from (33)
and (35), the behavior of each type of the density matrix given in (29) under the spatial
inversion and time reversal depends not only on the structure of the γ-matrix, but also on
the angular momentum of the spherical harmonics. Let us consider the case of an atom. We
treat the group G = O(3)× P × T . Here, we introduce the subgroup of G as
10
O(3)× P × T, O(3)× P, O(3)× T,
P × T, O(3), P, T, 1. (36)
For example, O(3)× P × T -invariant solution is given as
QS∗ = QS, QV ∗0 = QV
0 , others = 0. (37)
There is no case for O(3)× P -invariant solution. O(3)× T -invariant solution is given as
QS∗ = QS, QV ∗0 = QV
0 , QA∗0 = QA
0 , others = 0. (38)
O(3)-invariant solution is given as
QS∗ = QS, QV ∗0 = QV
0 , QA∗0 = QA
0 , QP∗ = QP , others = 0. (39)
Due to the rotational symmetry, each density matrix can only have an s-wave component,
in all cases (37)∼(39). All solutions given above are for closed shell states. We may have
magnetic QED-HF solutions for systems under discussions, if time reversal symmetry is
broken ( For example, the cases of (38) and (39) ).
B. Nonrelativistic limit
To consider the nonrelativistic limit, we take the standard representation. The four
component spinor ψ is partitioned as [17]
ψ =
φ
χ
, (40)
where φ is the large component and χ is the small component. Then,
QS =1
4trQ =
1
4〈ψψ〉 =
1
4〈φ†φ− χ†χ〉, (41)
QV0 =
1
4trγ0Q =
1
4〈ψγ0ψ〉 =
1
4〈φ†φ+ χ†χ〉, (42)
QA0 =
1
4trγ5γ
0Q =1
4〈ψγ5γ
0ψ〉 = −1
4〈χ†φ+ φ†χ〉, (43)
QP =1
4triγ5Q =
1
4〈ψiγ5ψ〉 =
1
4〈−iχ†φ+ iφ†χ〉. (44)
11
Therfore, at the nonrelativistic limit, the QS and QV0 coincide with each other, while the
QA0 and QP vanish: Note that, on the other hand, QS and QP vanish at the ultrarelativistic
limit.
Under the presence of the vectorial density matrices like the QVi , Q
Ai and QT
µν , the O(3)
rotational symmetry will be broken. For P × T -, P -, T - and 1 (no symmetry)-invariant
solutions, simple classification is impossible, because there are various possibilities of the
angular momentum dependences of the Q. This situation demands us futher investigation
in more detail in the future. It must be noted that we have to solve QED-HF equations
to find which type of the density matrices ( and solutions ) will be realized. The natural
orbital analysis of a resulted solution is useful to elucidate the type and magnitude of broken
symmetry.
IV. POST QED-HF METHODS
A. QED-CASSCF and CASPT2 methods
The standard approach for inclusion of fluctuation from the HF mean field approximation
is the MBPT approach [6]. For the purpose, the total Hamiltonian in (9) is devided into
the zeroth-order and interaction parts as
H0 = HK + H(HF )I , (45)
Hint = HI − H(HF )I , (46)
where H(HF )I denotes the HF mean field interaction potential. The remaining interaction
Hint is regarded as the perturbation to the HF solution in the case of the MBPT approach [6].
However, such single-reference (SR) MBPT approach often breaks down in the case of
near degenerate ground states, for which generalized QED-HF solutions exist, because of
the HF instability in (27) [11]. To consider these problems, we introduce the density matrix
which is defined as
12
ρ++ij (κ) ρ+−
ij (κ)
ρ−+ij (κ) ρ−−
ij (κ)
=
〈Φ(κ)|a†jai|Φ(κ)〉 〈Φ(κ)|bjai|Φ(κ)〉
〈Φ(κ)|a†j b†i |Φ(κ)〉 〈Φ(κ)|bj b
†i |Φ(κ)〉
, (47)
where we write (α, β, β∗, γ) as (κ). Then we diagonalize the density matrix of the most
stable QED-HF solution for systems under consideration [14], we obtain the natural orbital
given by four-component bispinor η(±)i and mi as its occupation number. By using the
occupation number mi, we can select the active space for treatments of the near degeneracy
effects [18]. Then the trial CI wavefunction for the QED-MCSCF are easily constructed.
The QED-MCSCF wavefunction is obtained like the nonrelativistic case:
|Ψ0〉 =∑
I
C(0)I eiS1 |ΦI〉. (48)
Here |ΦI〉 is the configuration state function ( CSF ), CI is the CI coefficient, and eiS1 is
the orbital rotation operator given in previous section. With introducing the orthogonal
complement to (48), |ΨK〉 =∑
J C(K)J |ΦJ〉, the variational operator of the MC function is
given as
T =∑
K 6=0
TK{|ΨK〉〈Ψ0|+ |Ψ0〉〈ΨK |}, (49)
where, T † = T and S†1 = S1 will be satisfied. Then we can write the transformed energy as
E = 〈Ψ0|e−iT e−iS1HeiS1eiT |Ψ0〉. (50)
This expression is, at a glance, same as the nonrelativistic case. Only the definition of
the orbital rotation operator is different. From (50), we can formally derive the variational
condition such as the generalized Brillouin theorem in our QED theory. Therefore the multi-
reference ( MR ) MBPT approach based on QED-MCSCF may be feasible like CASPT2 [19]
for the nonrelativistic case.
B. QED-CC and MRCC methods
The coupled-cluster wave function [20∼22] can also be introduced to obtain dynamical
correlation corrections, by obeying the exponential ansatz:
13
|ΨCC〉 = eiS|Φ0〉. (51)
Here the operator S is given as
S =∑
n
Sn = S1 + S2 + · · ·+ SN + · · · . (52)
n denotes the excitation level. |Φ0〉 is the QED-HF state. In the relativistic theory, the
expansion given above can not be truncated at N-excitation level, because of the presence
of the Dirac sea. Here, the excitation operator S is approximately truncated in the second
order; S ∼= S1 + S2. Even under this approximation, S2 includes the operators given as
follows:
a†i aja†kal, a†i aj a
†k b
†l , a†i aj bkal, a†i aj bk b
†l , a†i b
†j a
†k b
†l ,
a†i b†j bkal, a†i b
†j bk b
†l , biaj bkal, biaj bk b
†l , bib
†j bk b
†l . (53)
With considering that the operators a†a, a†b†, ba, bb† in exp(iS1) form the Lie algebra,
there is no other posibility. To write the exponential form, the excitation operators have
to be bosonic operators. Thus we have to devide the one-particle function space to two;
occupied and unoccupied spaces. Quantum number i, j, k, l will satisfy some conditions
like the nonrelativistic case. These operators have to take the form which will give two
particle excitation with respect to HF state. The operators given above include redundant
operators, which will not give the excitation with operating the HF state. The Schrodinger
equation is given as
HeiS|Φ0〉 = EeiS|Φ0〉. (54)
Then the projected coupled-cluster equations are given as
〈Φ0|HeiS|Φ0〉 = E0, (55)
〈Φ0|SnHeiS|Φ0〉 = E0〈Φ0|Sne
iS|Φ0〉. (56)
If we replace |Φ0〉 in (51) with |Ψ0〉 in (48), we may have the MRCC formulation [14,23,24]:
14
|ΨMRCC〉 = eiS|Ψ0〉 ∼= ei(S1+S2)|Ψ0〉. (57)
However, detailed formulations are rather complex because of redandancy. They will be
discussed elsewhere. A Schema of this work given in Fig. 2.
V. DISCUSSION
In this paper, we have developed our method which combines QED with many-body
techniques. We have given the QED Hamiltonian written by creation-annhilation opetrators,
and introduced the relativistic Slater determinant in the Thouless form. We have performed
the group-theoretical classification of the density matrix. After these preparations, we have
discussed the possibilities of the MCSCF and coupled-cluster method in QED.
Now we discuss the relation between our theory and other relativistic theories.
In the context of the electronic structure of atoms, there are three effects in atoms: The
electron correlation, the relativistic effect and the QED effect [25]. The electronic structure
of atoms is determined by the relation of these three factors. The electron correlation
depends on the electron numbers and can be treated by several many-body techniques. The
relativistic effect becomes large with increasing the atomic number. The relativistic effect in
neutral or almost neutral heavy elements can be treated satisfactorily by the Dirac-Coulomb-
Breit no-sea scheme [6], as discussed in introduction.
In fact, our QED Hamiltonian with neglecting positron states and adopting only the
density-density interaction taken into account, will derive the Dirac-Coulomb Hamilto-
nian [26,27] ( as illustrated in Fig. 3 ). Futher, we can have the Dirac-Coulomb-Breit
Hamiltonian with adding the Breit operator. Moreover, if the small component χ is approx-
imated by using the large component φ, and take into account the fact that χ gives small
contribution, then the Dirac equation can be reduced to the Schrodinger-Pauli equation [28],
for which 2-component spinor φ ( GSO ) will correctly allow for relativistic effects up to
(v/c)2. These terms will be calculated by ab initio GSO program package [4,5].
15
On the other hand, the case to describe the inner core electrons of heavy elements or,
electrons of highly ionized heavy elements such as lithium-like uranium, the QED effect can
not be neglected and we must take the Dirac sea into account. This effect can be treated
by the perturbation theory in QED [9]. But this method can only take into account the
one-particle QED correction. Practically this method treat the system of atoms with a few
electrons. For an illminative example of the QED effects, Fig. 4 shows the collision of two
uranium atoms [29]. This collision generates the pair-creations, and the QED effects might
be appeared clearly. Moreover, the near orbital degeneracy effects will become important in
this case, and also in clusters of these atoms or ions.
In the case of heavy atoms, as the ionicity becomes high and the electron number de-
crease, the many-body effect becomes small and the QED effect becomes large. Therefore
we propose that our theory should be applied to the cases where both the many-body effects
and the QED effects can not be neglected. Heavy elements in middle level of ionicity should
be one of the subject for our theory. The Dirac-Coulom-Breit no-sea scheme has been ap-
plied to only neutral or almost neutral atoms. The method of the QED correction works
only for the highly ionized heavy atoms. Our theory should be suitable for intermediate case
between them. The collision of two uranium atoms is also one of the interesting subject of
our theory. Recently, experiments of x-ray irradiation to cluster plasma are performed, and
verious new phenomena were studied [30]. The cluster of heavy ions under middle revel of
ionicity is now an interesting subject in this area. The importance of the relativistic and
QED effects is discussed in such objects. Fig. 5 illustrate scope and limitation of these the-
ories. It is noteworthy that many interesting phenomena appear in the intermediate regime
even in the nonrelativistic case [31]. An application of the QED scheme to treat the strong
electron-electron interaction in solid state physics is also discussed in relation to spin density
wave and charge density wave states of mixed valence ( MV ) systems [32].
16
ACKNOWLEDGMENTS
The authors wish to thank Dr. D. Yamaki and Dr. S. Yamanaka for their helpfull
discussions, and Mr. Y. Kitagawa for his help to depict the figures. The authors are
grateful for the financial support of a Graint-in-aid for Scientific Reserch on Priority Areas
( Nos. 10149105 and 11224209 ) from Ministry of Education, Science, Sports and Culture,
Japan.
17
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20
+m
-m
Fig. 1
εF
0
Dirac Sea
QED Hamiltonian
neglect positron states
only take into account
the density-density
interaction
Dirac-Coulomb Hamiltonian
Fig.2
QED Hamiltonian written by
creation-annihilation operator
Slater determinant
of Thouless form
RPA
Fig.3
Stability Condition
QED CASSCF QED GHF
QED MRCC
ρ
1.02MeV
negative energy
electron
positron
within association
1.02MeV
positron
1s
2p
electron
U ion U ion
(A) (B)Fig.4
before association
this work
ionicity
non-relativistictheory
atomic number
Dirac-Coulomb-Breit
one-particle QED collection
Fig.5
20
40
15
60
100
100